Have you ever stopped to think about whose spirit you're feeling when you "feel the Spirit"? The question is if you're feeling the influence of our Heavenly Father's spirit, Christ's spirit, or the spirit of the Holy Ghost/Holy Spirit. Since we maintain that they are physically separate we also maintain that they have separate spirits. Can we, or should we, even figure out whether we're feeling the unique influence of either of the three?
After posing this question and thinking about it for some time, I've decided that perhaps it's not wise to try and subdivide the influence of the Godhead. Perhaps us Mormons are already too guilty of trying to separate the Godhead too much whenever we emphasize their "threeness" more than their "oneness". They are infinitely more one than they are separate.
I'm content with the idea that I can feel the united and interconnected energy/influence/spirit of the one God/Godhead we worship. With that in mind, I now perceive the phrase "the Spirit of God" differently. I like thinking of it more as the Spirit of the one God/Godhead. By any means, it would be foolish to try and limit either of their influence on us.
During the sacrament prayer, in exchange for our promises to follow and remember Christ, we're “promised that his Spirit, meaning the Spirit of Christ, will always be with [us]. This is no small matter, because the Spirit of Christ is the Light that radiates from God to fill the immensity of space and uphold all of creation. It is the light that enlightens the eye and the light that enlightens the understanding. ‘The Glory of God is intelligence, another scripture says, and this great light—intelligence can flow into humble communicants through the covenant of the sacrament prayer (D&C 93:36)". (Richard Bushman, "A Very Short Introduction to Mormonism").
I also like how Blake Ostler described a loving interpenetration of freely cooperating wills. He once wrote: "I assert that both the Father and the Son are eternally divine. However, there is a priority of the Father in the sense that the Father offers his love to the Son, and in each moment of eternity the Son has freely chosen to fully return that love. They both offer their love to the Holy Ghost and the Holy Ghost has freely chosen in each moment of eternity to return that love. "It is in virtue of this loving interpenetration of freely cooperating wills that these three are one God and also have been eternally one God. Now they are inviting us into this same relationship."
So in short, the question of "whose spirit are your feeling?", while interesting, perhaps isn't as important as some might think. Instead of trying to understand which spirit we're feeling (whether that of the Father, Son, or Holy Ghost), we ought to recognize what our scriptures assert--that they're "one God". And perhaps we're never more "at one" with them than when we're filled their love.